Recently, the linearizer for a high-power amplifier has been required to operate with precision because of the development of a high bit rate communication system. Problems arise when RF (radio frequency) signals are passing through the nonlinear high-power amplifier (HPA). Often, to increase the efficiency of high-power amplifiers (HPAs), they are driven into a nonlinear region.
The effect of nonlinear HPAs in digital communication systems has been investigated in various papers [non-patent documents 1-4]. The results show that the HPA nonlinearity generates in-band and out-of-band signal distortions.
Digital predistortion (PD) is one of the most promising techniques to reduce out-of-band and in-band distortions by linearizing distortion on a signal which passes through nonlinear HPAs. The basis of the predistortion approach for a compensating nonlinear distortion HPA is to design the inverse system of the compensated nonlinear system (HPA).
FIG. 1 illustrates the simplified block diagram of a transmitter with the adaptive digital predistorter (PD) with indirect learning architecture according to a prior art.
A signal x(n) is a digital baseband signal sent from a signal transmitter (not illustrated). The signal x(n) is input to the predistorter 10. The predistorter 10 distorts the input signal x(n) according to a memory polynomial which represents an inverse characteristic of HPA 12. The predistorter 10 may be a calculator with hardware logics, a digital signal processor (DSP), or a processor which operates with an installed program. The output signal z(n) of the predistorter 10 is input to an RF I-Q modulator 11. The RF I-Q modulator 11 modulates the baseband signal x(n) from the predistorter 10 into a radio frequency signal.
The output signal from RF I-Q modulator 11 is input to HPA 12. HPA 12 amplifies the output signal from the RF I-Q modulator 11 with a gain G. The output signal y(n) from HPA 12 is sent out from an antenna 13. Additionally, the signal y(n) is input to an attenuator 14 with a gain 1/G through a feedback path, where a gain G applied to a signal by HPA 12 is cancelled. The output signal from the attenuator 14 is input to RF I-Q demodulator 15 where modulation applied to a signal by RF I-Q modulator 11 is cancelled. The output signal from RF I-Q demodulator 15 is digitized by an analog-to-digital converter (ADC) 18. The output signal from ADC 18 is passed through a low pass filter 16, outputting a signal yFB.
The signal yFB is input to a training predistorter 17. The training predistorter 17 receives an error signal ε which is obtained by subtracting the signal z^(n) from the signal z(n). The training predistorter 17 calculates coefficients A=[a10, . . . , aK0, . . . , a1Q, . . . , aKQ]T of the memory polynomial so that the error signal ε is minimized as described later. The memory polynomial with the calculated coefficients, which models an inverse characteristic of HPA 12, is copied to the predistorter 10.
The HPA nonlinearity causes the appearance of parasitic frequency components in the output signal y(n), occupying P times the bandwidth of the input signal x(n). It is normal practice to assume P to be equal to 5 [non-patent document 3]. As a general rule of thumb, the 5th order of HPA nonlinearity causes the appearance of parasitic components occupying 5 times the bandwidth of the input signal. In adaptive PD (PreDistorter) systems, it is necessary to have a feedback path. Input signal x(n) (FIG. 1) from the DAC (digital-to-analog converter, not illustrated) output can have the sampling clock as high as 1 GHz for signals with bandwidth of 100 MHz. Accounting for intermodulation products (known as IMD in the relevant art field) in an HPA output, the feedback bandwidth for a sample-by-sample comparison of the signals z(n) and z^(n) can reach 5*100=500 MHz.
In such cases, system costs are largely driven by high-performance (hi-speed and hi-bit-resolution) analog-to-digital converters (ADCs). An example of ADC is PXI-5922 sold by National Instruments, which may be used as a high-speed ADC. A catalog of digitizers is available from a web site (www.ni.com). For example, digitizing of the baseband signal with a bandwidth of 500 MHz requires an ADC sampling rate equal to at least 1 GHz according to the Nyquist sampling theorem [non-patent document 2].
However a higher sampling rate of 2 GHz is preferable for practical implementations. A less costly alternative adapts a DPD (Digital PreDistorter) using only narrowband feedback.
For baseband predistortion systems, the transmission path includes the PD. In adaptive PD systems illustrated in FIG. 1, it is necessary to add a feedback path from HPA 12, an attenuator 14 with gain 1/G, and a baseband PD adaptation system (Training PD block 17). The actual PD 10 placed in the transmission path is an exact copy of the training PD 17 in the feedback path.
In most practical implementations, the feedback path bandwidth is not infinite; it has a finite value. As illustrated in FIG. 1, the PD's feedback path bandwidth is limited by the pass-band of the anti-aliasing LPF 16. The aliasing on the signal occurs due to ADC 18. The analog signal has a broad bandwidth and when the analog signal is digitized by ADC 18, because a sampling clock frequency of ADC 18 is limited, the digitalizing process inevitably induces out-of-band frequency components, which are aliasing. Generally speaking, the ability of a PD to linearize deteriorates with a narrow feedback path bandwidth, as a narrowband feedback path bandwidth introduces additional distortions into the signal coming from the HPA output.
FIG. 2 illustrates a spectrum in the HPA equipped with the PD in the case in which the anti-aliasing LPF is placed into the PD feedback path and with the normalized LPF pass band as a parameter. FIG. 3 illustrates the LPF transfer functions corresponding to FIG. 2.
The indexes with LPF define the normalized LPF pass band bandwidth—the LPF pass band bandwidth Δf normalized to the PD's input signal x(n) bandwidth ΔF. The baseband input x(n) is an OFDM signal with 2048 subcarriers and RF bandwidth is ΔF=20 MHz.
In FIGS. 2 and 3, a horizontal axis represents a frequency offset from a center of the signal band in frequency and a vertical axis represents an intensity of frequency components with an intensity of an in-band signal as 0 in dB. “OBO” is an abbreviation for Output Back-Off, which means the ratio of the signal power measured at the output of a high-power amplifier to the maximum output signal power. The numbers after LPF are normalized bandwidths. “Mem” means “memory” and Mem LPF means LPF with memory.
As can be seen from FIGS. 2 and 3, inserting the LPF into the PD's feedback path significantly deteriorates the linearization performances and causes the HPA output signal spectrum regrowth in an out-of-band frequency. That is, the lesser the out-of-band frequency components, the better the linearization performances of PD.
The operation of the memory polynomial predistorter is described below.
Firstly we briefly review memory polynomial linearization, following the treatment in [non-patent document 3].
FIG. 1 illustrates the indirect learning architecture that is used for PD identification. Here, PD identification means making the memory polynomial of the training predistorter 17 identical to the memory polynomial of the predistorter 10. The objective of the linearizer is to find a transformation of the signal (z(n)=HPA−1(x(n)) that in tandem combination with nonlinear HPA suffered with a memory effect will result in an identity system that produces the signal of interest without distortions at the output of HPA, i.e., demodulated (y(n))=x(n). In this approach, two identical memory-polynomial systems (denoted in FIG. 1 as PD 10 and the training PD 17) are used for training predistortion.
The feedback path comprises from the LPF 16 that limits the feedback path bandwidth and a training PD 17 that has yFB(n) as its input and z^(n) as its output. The actual predistorter 10 is an exact copy of the training predistorter 17; it has x(n) as its input and z(n) as its output.
The LPF 16 is a linear system with a memory, and therefore it introduces the interference (ISI; Interference Signal Indicator) between feedback samples yFB (n) and the HPA output. Thus the signal samples in the feedback path at the LPF output become correlated with themselves. For further PD performance degradation analysis, it is possible to replace the LPF 16 in PD illustrated in FIG. 1 with some equivalent external interference source as illustrated in FIG. 4.
FIG. 4 illustrates the linearizer with an LPF equivalent interference source.
In FIG. 4, like components to those of FIG. 1 are denoted with like numerals.
In FIG. 4, ADC 18, RF I-Q modulator 11, and RF I-Q demodulator 15 in FIG. 1 are omitted for brevity of explanation and the source 20 of the interference δ(n) which is added to the output of the attenuator 14 is introduced as an equivalent structure of LPF 16 in that LPF 16 introduces the interference.
For the moment, let's assume that the bandwidth of the LPF placed in the PD feedback path is wide enough—i.e., that such an LPF does not introduce any significant distortions into the PD's feedback signal y(n) from an HPA output. The exact influence of the LPF placed in the feedback path on the PD linearization performances will be discussed later. Thus, the following approximation becomes valid: yFB(n)=y(n) and ISI term δ(n)=0. The memory polynomial predistorter can be described in a similar manner to [non-patent document 3] as:
                              z          ⁡                      (            n            )                          =                              ∑                          k              =              1                        K                    ⁢                                    ∑                              q                =                0                            Q                        ⁢                                          q                kq                            ·                              x                ⁡                                  (                                      n                    -                    q                                    )                                            ·                                                                                      x                    ⁡                                          (                                              n                        -                        q                                            )                                                                                                          k                  -                  1                                                                                        (        1        )            where akq are the unknown complex memory-polynomial predistorter coefficients; x(n) is transmitting signal samples at the discrete time n. The memory polynomial predistorter calculates the output signal z(n) using expression (1) which is preinstalled in the memory polynomial predistorter, of which akq is variable; k is an index indicating an order of the memory-polynomial; K is the order of the memory-polynomial; q is an index indicating a memory effect of the memory-polynomial; and Q indicates a maximum sample included in the memory-polynomial as the memory effect.
Ideally, after initialization through indirect learning, at convergence, we should have z(n)=z^(n), or in the matrix form [see non-patent document 1]Z=Y·A   (2)where, ykq(n)=y(n−q)·|y(n−q)|k; Z=[z(0),z(1), . . . , z(N−1)]T, Y=[Y10, . . . , Yk0, . . . , YKQ], Ykq=[ykq,(0) . . . , ykq(N−1)]T, A=[a10, . . . , aK0 . . . , a1Q, . . . , aKQ]T, and N is the number of available data samples n.where Z, Y and A are matrix form of z(n). y(n) and akq, respectively. And Here, it is assumed that y(n) equals z(n) at convergence.
The error term does not exist explicitly in the expression (2) for the indirect learning. Therefore, for further discussion, in order to investigate the influence of the linearization error, we employ the linear regression approach.
Indirect learning and polynomial regression are explained below.
The polynomial regression fits a nonlinear relationship between the value of yFB(n) and the corresponding conditional mean of z^(n) (see FIG. 1). Since z^(n) is linear with respect to unknown parameter akq, i.e., the akq in expression (1), these unknowns can be estimated by a simple least-square method, by treating yFB, yFB2, . . . yFBK−1 as being distinct independent variables in a multiple regression model. Because usually only the approximate solution for polynomial regression can be obtained, there is an unobserved random ε(n) error (a.k.a. disturbance term) with a mean zero in expression (3) below. Thus,ε(n)=z(n)−z^(n)   (3)
The regression results were less than satisfactory when the independent variables were correlated and the disturbance term is not i.i.d. (Independent and Identically Distributed). For the relatively wideband LPF, we can assume that the ε(n) error is an i.i.d. random variable. Given y(n) and z(n), a PD training task is to find parameters akq of memory-polynomial PD as expressed in expression (2), which yields the predistorter. The algorithm converges when the error energy ∥ε(n)∥ is minimized. Thus, in the polynomial regression model we have:
                              z          ⁡                      (            n            )                          =                ⁢                                            z              ^                              (                n                )                                      +                          ɛ              ⁡                              (                n                )                                              =                    ⁢                                                    ∑                                  k                  =                  1                                K                            ⁢                                                ∑                                      q                    =                    0                                    Q                                ⁢                                                      a                    kq                                    ·                                                            y                      FB                                        ⁡                                          (                                              n                        -                        q                                            )                                                        ·                                                                                                                                    y                          FB                                                ⁡                                                  (                                                      n                            -                            q                                                    )                                                                                                                                  k                      -                      1                                                                                            +                          ɛ              ⁡                              (                n                )                                                                        (        4        )            Our goal is to minimizeΣε2(n)over all n=0, . . . , N−1 available data samples. With the assumption that the LPF pass band bandwidth can be rewritten as a linear regression function in terms of the unknown regression coefficients akq, the transfer function between input and output of the feedback path including LPF and the training PD is expressed as:
                              z          ⁡                      (            n            )                          =                                            ∑                              k                =                1                            K                        ⁢                                          ∑                                  q                  =                  0                                Q                            ⁢                                                a                  kq                                ·                                                      y                    kq                                    ⁡                                      (                    n                    )                                                                                +                      ɛ            ⁡                          (              n              )                                                          (        5        )            
Equation (5) can be expressed in matrix form in terms of a design matrix Y, a response vector Z, a parameter vector A, and a “disturbance term” vector ε of random i.i.d. errors that add noise to the linear relationship between the dependent variable and regressors, which when using pure matrix notation, is written asZ=Y·A+ε  (6)Where design matrix Y and parameter matrix A have been defined in expression (3) andε=[ε(0),ε(1), . . . , ε(N−1)]T is the disturbance vector. A similar expression, however, with an implicit disturbance term, has also been reported in [non-patent document 1]. The vector of an estimated polynomial regression coefficient akq using a least-squares solution for expression (6) is:A=(YT·Y)−1·YT·Z   (7)
For the linear regression analysis described above, ISI δ(n) increases the disturbance term in expressions (4) and (6) from ε(n) to ε(n)+δ(n). Therefore, the total noise power for the disturbance term due to LPF inclusion into the PD's feedback path can be calculated as a sum of the partial noise powers for δ^(n) and δ(n):σΣ2=σε2+σδ2   (8)where σ means standard deviation of respective noise components; Σ means total noise; ε means ε(n)'s noise components; and δ means δ(n)'s noise components.
An increase in the disturbance term in expressions (4) and (6) results in the increase in the spectral regrowth as it is illustrated in FIG. 2. If the spectral regrowth is increased, the training PD's performance to learn the characteristic of HPA degrades, thereby degrading the linearization performance of the linearizer.
Some of the prior arts describe a predistorter using an amplifier replica, a system in which a distortion is applied to an input signal in advance, and a predistorter using a distortion compensation polynomial.